Question

Find the quadratic polynomial whose zeros are 3 and 2.

Answer 1

Hint: Assume the given two zeros of the quadratic polynomial as $\alpha \text{ and }\beta $. Write these zeros in the form of factors of the quadratic polynomial. At last multiply the two factors to get the required quadratic polynomial.

Complete step-by-step answer:

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2 or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree. For example: $f(x)=a{{x}^{2}}+bx+c$, where $a\ne 0$, is a quadratic polynomial. The graph of a quadratic polynomial is a parabola. If the quadratic polynomial is set equal to zero, then the result is a quadratic equation. The solution of the quadratic equation is known as its roots or zeros. A quadratic polynomial can have a maximum of two roots.
Now, we come to the question. It is given that 2 and 3 are the zeros of the given polynomial. Therefore, $(x-2)\text{ and (}x-3)$ will be the factors of the polynomial. Now, suppose we have to find a number whose factors are given to us, how will we find the number? The solution to this problem is that we have to multiply all the factors given, to get the number. Similarly, we have factors of the quadratic polynomial and we have to find the polynomial. So, to get the polynomial $p(x)$, multiply the factors. Therefore,
$\begin{align}
  & p(x)=\left( x-2 \right)\left( x-3 \right) \\
 & \text{ }={{x}^{2}}-3x-2x+6 \\
 & \text{ }={{x}^{2}}-5x+6 \\
\end{align}$

Note: There is an alternative or shortcut method to find the polynomial, if its zeros are given. Assume that the given roots are: $\alpha =3\text{ and }\beta =2$. The polynomial is given by the formula: $p(x)={{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta $. The polynomial obtained will be the same as we have found in the above solution.